The suicide attack in Nazran, Ingushetia's main city, injured more than 138 people, including children.
The republic borders Chechnya and has seen a spate of shootings, bombings and other attacks on police and government.
Ingushetia's leader blamed militants, but Russian President Dmitry Medvedev sacked the region's interior minister, saying the attack had been preventable.
"This act of terror could have been averted," Mr Medvedev said. "The police must protect the people and the police must also be able to defend themselves."
Monday's bomb attack was the deadliest strike in Ingushetia since 2004, when some 90 people were killed when hundreds of gunmen staged a raid on the centre of Nazran.
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